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Properties

Label 459186.f
Number of curves $6$
Conductor $459186$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("f1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 459186.f have rank \(1\).

Complex multiplication

The elliptic curves in class 459186.f do not have complex multiplication.

Modular form 459186.2.a.f

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + 2 q^{10} + 4 q^{11} - q^{12} + q^{13} - q^{14} + 2 q^{15} + q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 459186.f

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
459186.f1 459186f6 \([1, 1, 0, -158931356, -771232469154]\) \(746387736968472436657/28813240527162\) \(17138787419138291545002\) \([2]\) \(86016000\) \(3.3512\)  
459186.f2 459186f4 \([1, 1, 0, -10402346, -10853055360]\) \(209280938961918097/35633951258724\) \(21195905228066339902404\) \([2, 2]\) \(43008000\) \(3.0046\)  
459186.f3 459186f2 \([1, 1, 0, -2984726, 1823657220]\) \(4943668694576977/442262240784\) \(263067894816040523664\) \([2, 2]\) \(21504000\) \(2.6581\)  
459186.f4 459186f1 \([1, 1, 0, -2917446, 1916786196]\) \(4616835877167697/42561792\) \(25316746465151232\) \([2]\) \(10752000\) \(2.3115\) \(\Gamma_0(N)\)-optimal*
459186.f5 459186f3 \([1, 1, 0, 3356414, 8541460936]\) \(7030109576481263/57036871916388\) \(-33926861572757544484548\) \([2]\) \(43008000\) \(3.0046\)  
459186.f6 459186f5 \([1, 1, 0, 19444744, -61634894286]\) \(1366917738131980943/3550749039197178\) \(-2112068335532824591788138\) \([2]\) \(86016000\) \(3.3512\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 459186.f1.